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5.1 Introduction to Normal Distributions and the Standard Normal Distribution

5.1 Introduction to Normal Distributions and the Standard Normal Distribution. Interpret graphs of normal probability distributions Find areas under the standard normal curve. Normal distribtuion.

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5.1 Introduction to Normal Distributions and the Standard Normal Distribution

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  1. 5.1 Introduction to Normal Distributions and the Standard Normal Distribution Interpret graphs of normal probability distributions Find areas under the standard normal curve

  2. Normal distribtuion A normal distribution is a continuous probability distribution for a random variable x where the mean, median, and mode are equal.

  3. Normal Curve The graph of a normal distribution is called the normal curve.

  4. A normal distribution has the following properties: The mean, median, and mode are equal. The normal curve is bell-shaped and is symmetric about the mean. The total area under the normal curve is equal to 1. The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean. Between μ-σ and μ+σ (in the center of the curve), the graph curves downward. The graph curves upward to the left of μ-σ and to the right of μ+σ. The points at which the curve changes from curving upward to curving downward are called inflection points.

  5. Try it yourself 1 • Understanding Mean and Standard Deviation Consider the normal curves shown at the right. Which normal curve has the greatest mean? Which normal curve has the greatest standard deviation? B has the greatest mean. Curve C is more spread out, so curve C has the greatest standard deviation.

  6. Try it yourself 2 • Interpreting Graphs of Normal Distributions The scaled test scores for the New York State Grade 8 English Language Arts Test are normally distributed. The normal curve shown below represents this distribution. What is the mean test score? Estimate the standard deviation of this normal distribution. Mean: 660 Standard Deviation: 30

  7. Standard normal distribution The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

  8. Properties of the Standard Normal Distribution

  9. Try it yourself 3 • Using the Standard Normal Table • Find the cumulative area that corresponds to a z-score of -2.19. • Find the cumulative area that corresponds to a z-score of 2.17. 0.0143 0.0143

  10. Try it yourself 4 • Finding Area Under the Standard Normal Curve Find the area under the standard normal curve to the left of z = 2.13. 0.9834

  11. Try it yourself 5 • Finding Area Under the Standard Normal Curve Find the area under the standard normal curve to the right of z = -2.16. 0.9846

  12. Try it yourself 6 • Finding Area Under the Standard Normal Curve Find the area under the standard normal curve between z = -2.165 and z = -1.35. 0.0733

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